This dissertation presents some results on the application of low energy effective field theory vortex dynamics in condensed matter and materials systems. For the first half of the presentation we discuss the possibility of non-Abelian gapless excitations appearing on U(1)
vortices in the B phase of superfluid ^3
He. Specifically, we focus on superfluid ^3
He-like systems with an enhanced SO(3)L
rotational symmetry allowing for non-Abelian excitations to exist in the gapless spectrum of vortices. We consider a variety of vortices in the B-phase with different levels of symmetry breaking in the vortex core, and show conditions on the phenomenological parameters for certain vortices to be stable in the bulk. We then proceed to develope the low energy effective field theory of the various vortex types and consider the quantization of excitations. The process of quantization leads to interesting surprises due to non-lorentz symmetry that are not typically encountered in the analogous cases of U(1)×SU(N)
gauge models discussed in high energy theory.

The second half of this dissertation focuses on two types of vortices that appear in a particular model that is a modification of the well known Abelian-Higgs model. The specific modification includes a vector spin field in addition to the U(1)
Higgs field and gauge fields of the original model. The particular form of the lagrangian results in a cholesteric vacuum structure, with interesting consequences for the vortices in the model. We observe the effects of such a modification on the well known U(1)
vortex appearing in the original model due to the emergent spin field in the vortex core. We also consider a new type of vortex that is most closely related to a spin vortex. This vortex appears due to the topology introduced by the new spin field. The low energy effective field theory is also investigated for this type of vortex.